medium difficulty7 min

Integration by Partial Fractions

Integration by partial fractions decomposes a proper rational function into simpler fractions whose integrals are standard, usually logarithmic or inverse trigonometric.

In one line: Break the rational expression first; integrate only after the pieces are simple.

Practice This Concept

Understand it

This method applies to rational functions where the numerator degree is less than the denominator degree after any necessary division. The denominator is factorised, unknown constants are found, and each simpler fraction is integrated separately.

Example

1/[(x-1)(x+2)] can be written as A/(x-1)+B/(x+2), then A and B are found before integrating.

Formula and rule

Method condition: use partial fractions for rational functions P(x)/Q(x), after ensuring degree P < degree Q.

For distinct linear factors, write A/(x-a)+B/(x-b).

For repeated factors, include A/(x-a)+B/(x-a)² and so on.

Then integrate ∫dx/(x-a)=ln|x-a|+C.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Evaluate ∫dx/[(x+1)(x+2)].

1Write 1/[(x+1)(x+2)]=A/(x+1)+B/(x+2).
2Then 1=A(x+2)+B(x+1).
3Put x=-1: 1=A, put x=-2: 1=-B, so B=-1.
4Integral =∫[1/(x+1)-1/(x+2)]dx=ln|x+1|-ln|x+2|+C=ln|(x+1)/(x+2)|+C.

Method you can reuse

1Check degrees of numerator and denominator.
2If needed, divide first.
3Factorise the denominator.
4Write the correct partial fraction form.
5Find constants by substitution or coefficient comparison.
6Integrate each term.
7Combine logarithms only if it simplifies cleanly.

Visual tip

No diagram is required. A factor table or coefficient comparison layout is useful for organizing constants.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Decompose 1/[(x+1)(x+2)] into partial fractions.

Think first, then show answer

1/[(x+1)(x+2)]=1/(x+1)-1/(x+2).

Common wrong answer

Writing A/(x+1)+B/(x+2) correctly but solving constants with sign errors, leading to ln|x+1|+ln|x+2| instead of a difference.

Don't mix this up

Students often skip checking whether the fraction is proper. If the numerator degree is greater than or equal to the denominator degree, polynomial division must come first.

Exam tip

After finding constants in the decomposition, recombine mentally or by substitution of easy x-values to check before integrating.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Usually appears as 3-mark or 5-mark questions with distinct linear factors, repeated linear factors, or a rational function needing division first.

1
Enough for 1 mark
Integration by partial fractions decomposes a proper rational function into simpler fractions whose integrals are standard, usually logarithmic or inverse trigonometric.
2
Add for 2 marks
Example: 1/[(x-1)(x+2)] can be written as A/(x-1)+B/(x+2), then A and B are found before integrating.
3
Add for 3 marks
This method applies to rational functions where the numerator degree is less than the denominator degree after any necessary division.

MCQ Quiz

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